Conventionally, various engineering applications use a model (hereinafter, “model formula”) expressing operation (output or next state) of an object to be modeled by a mathematical formula. The model formula includes input variables, state variables, physical constants, fitting parameters, etc.
The model formula is extracted by adjusting the fitting parameters (or physical constants) so that results of measurement and results of the model formula will match. Use of the model formula thus extracted makes it possible to easily predict the operation of the object to be modeled. It becomes possible, for example, to predict the operation at an input condition not yet measured from the model.
Conditions to be given to a simulation include preconditions and input conditions. A precondition refers to a condition that cannot be expressed without changing the fitting parameter when results of calculation are modeled by the model formula such as a condition not expressed by an input variable in a model formula. An input condition refers to a condition that can be expressed without changing the fitting parameter when the calculation results are modeled by the model formula, such as an input value and a value of the physical constant.
When conditions necessary for simulation are expressed, for example, by four variables of a, b, c, and d and when the model formula is expressed by a function of variables a and b, where a and b are input conditions and c and d are preconditions, if c or d changes, then a different model formula is necessary.
Since comprehensive simulation for various input conditions is very time-consuming, modeling is performed from the results of simulation in which some input conditions have been changed, making it possible to predict results of the input conditions not simulated (response surface methodology). One approach applies a polynomial model formula, etc., to calculation results or measurements of a part and predicting the whole by the least squares method (see, e.g., Japanese Laid-Open Patent Publication Nos. H6-195652 and 2002-353440).
With the conventional technologies described above, although partial calculation enables high-speed calculation, it can be extremely difficult to generate a model reflecting physical properties and therefore, it is a common practice to prepare the response surface by a model formula that does not have the physical properties such as a polynomial. Use of a model formula not having the physical properties causes various problems.
FIG. 6 is a diagram of examples of the model formula. (A) depicts a preferable model formula. (B) depicts a drastic error in the model formula when there are a small number of measured values. When the number of the measured values is small, a problem arises in that an impossible model formula is extracted or that the model formula is susceptible to measurement errors.
(C) depicts an error caused in the model formula by a low order of the model formula. When using a low-order polynomial, a problem arises in that the model formula is unable to adequately follow differences in operation, resulting in worsened accuracy.
(D) depicts an error caused in the model formula by a model formula of too high an order. In view of the low order in (C), when the order is raised to improve accuracy, a problem arises in that the model formula waves or in that measurement errors or calculation errors are introduced into the model formula. Thus, the generation of a model formula having physical properties is extremely difficult by the conventional technologies.